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Structural Design for Cold Region Engineering. Lecture 14 Thory of Plates Shunji Kanie. Theory of Plates Kirchhoff Plate. Kirchhoff Plate. Pure Bending. Such as Bernoulli Euler. Assumptions. Isotropic and homogeneous The thickness of the plate is thin

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Structural design for cold region engineering

Structural Design forCold Region Engineering

Lecture 14 Thory of Plates

Shunji Kanie


Theory of plates kirchhoff plate
Theory of PlatesKirchhoff Plate

Kirchhoff Plate

Pure Bending

Such as Bernoulli Euler

Assumptions

Isotropic and homogeneous

The thickness of the plate is thin

(Comparatively to the length and width )

Linear filaments of the plate

(Even after the deformation)

Kirchhoff hypothesis


Theory of plates kirchhoff plate1
Theory of PlatesKirchhoff Plate

Kirchhoff Plate

Length : a in x direction

Width : b in y direction

Thickness : h in z direction

Deflection

Rotation angle


Theory of plates kirchhoff plate2
Theory of PlatesKirchhoff Plate

Kirchhoff Plate

Rotation angle

Displacement due to deflection


Theory of plates kirchhoff plate3
Theory of PlatesKirchhoff Plate

Kirchhoff Plate

Stress-Strain Relation

Plane Stress !


Theory of plates kirchhoff plate4
Theory of PlatesKirchhoff Plate

Sectional Force

Bending Moments and Torsional Moment are calculated at least


Theory of plates kirchhoff plate5
Theory of PlatesKirchhoff Plate

X direction

y direction

z direction


Theory of plates kirchhoff plate6
Theory of PlatesKirchhoff Plate

z direction


Theory of plates kirchhoff plate7
Theory of PlatesKirchhoff Plate

Governing Equation


Theory of plates kirchhoff plate8
Theory of PlatesKirchhoff Plate

Governing Equation


Theory of plates kirchhoff plate9
Theory of PlatesKirchhoff Plate

Introducing Laplacian


Theory of plates kirchhoff plate10
Theory of PlatesKirchhoff Plate

Boundary Conditions

Simple support

Fixed support

Free support

Effective transverse shear

Kirchhoff force


Theory of plates solution
Theory of PlatesSolution

Simply Supported Plate

Assuming Deformation

Boundary Condition?


Theory of plates solution1
Theory of PlatesSolution

Simply Supported Plate

Governing Equation

Assumed Deformation


Theory of plates solution2
Theory of PlatesSolution

Simply Supported Plate

Assumed Deformation

Deformation


Theory of plates solution3
Theory of PlatesSolution

Simply Supported Plate

Deformation

Bending & Twisting Moments


Theory of plates solution4
Theory of PlatesSolution

Simply Supported Plate

Bending & Twisting Moments

If we are very LUCKY enough


Theory of plates solution5
Theory of PlatesSolution

Simply Supported Plate

If

qmn is successfully calculated and we can have the solution

Is there any good idea if q is uniformly distributed load?


Theory of plates solution6
Theory of PlatesSolution

Simply Supported Plate

Apply Double Fourier Expansion for q


Theory of plates solution7
Theory of PlatesSolution

Simply Supported Plate

When q is constant as q0

You can solve the problem for any shape of load distribution


Theory of plates solution8
Theory of PlatesSolution

Plate supported like

Assuming Deformation

Single Fourier Expansion


Theory of plates solution9
Theory of PlatesSolution

Plate supported like

If q is constant in x direction

m=1,3,5,…….


Theory of plates solution10
Theory of PlatesSolution

Plate supported like

If q is linear in x direction

m=1,2,3,4,…….


Theory of plates solution11
Theory of PlatesSolution

Plate supported like

If q is constant in x direction

m=1,3,5,…….

If q is linear in x direction

m=1,2,3,4,…….

Solve


Theory of plates solution12
Theory of PlatesSolution

Plate supported like

Solve

General Solution

Characteristic Equation


Theory of plates solution13
Theory of PlatesSolution

Plate supported like

General Solution

Singular Solution

should be constant such as


Theory of plates solution14
Theory of PlatesSolution

Plate supported like

General Solution

Singular Solution

Solution


Theory of plates solution15
Theory of PlatesSolution

Difference Method


Theory of plates solution16
Theory of PlatesSolution

Difference Method

Governing Equation

Simple support

Fixed support


Theory of plates solution17
Theory of PlatesSolution

Galerkin Method

Assume Approximation

Governing Equation

Weighted Residual

Same with Double Fourier


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